If you watch a forum (for new threads in my case), you will not receive any notifications about moved threads to that forum.
I guess it's a matter of what we consider New Threads. If it means only to new created threads, or just new threads in the forum.
It's a tricky issue. It's similar in concept to things that come up when merging threads, and I think we have that tagged as a future fix so I'm going to do the same here. The changes that I think are necessary are likely too substantial for an RC stage.
I think we'd need to do different calculations to determine who to notify than we normally do to prevent duplicate or unnecessary notify. Or maybe, we'd just need to create a whole new alert/email that explicitly said the thread was moved to a forum you watch.